Class of Quantum Many-Body States That Can Be Efficiently Simulated

We introduce the multiscale entanglement renormalization ansatz, a class of quantum many-body states on a $D$-dimensional lattice that can be efficiently simulated with a classical computer, in that the expectation value of local observables can be computed exactly and efficiently. The multiscale entanglement renormalization ansatz is equivalent to a quantum circuit of logarithmic depth that has a very characteristic causal structure. It is also the ansatz underlying entanglement renormalization, a novel coarse-graining scheme for many-body quantum systems on a lattice.

Figure 1

Example of quantum circuit $\mathcal{C}$ that transforms the product state $|0{⟩}^{\otimes N}$ into the state $|\Psi ⟩$ of lattice $\mathcal{L}$, case $D=1$. This circuit contains $2N-1$ gates organized in $O(\log N)$ layers that are labeled by a discrete time $\theta$. Notice the periodic boundary conditions.

Figure 2

The MERA $\mathcal{M}$ is made of three types of tensors: $t$, $w$, and $u$; see Eqs. (1, 2). Its causal structure is inherited from that of the circuit $\mathcal{C}$: The causal cone ${\mathcal{C}}^{[s]}$ for site $s$ has bounded width. When reversing the arrow of time $\theta$, $\mathcal{M}$ corresponds to $O(\log N)$ entanglement renormalization transformations labeled by $\tau$.

Figure 3

Detail of a causal cone of a simple MERA construction for a square lattice, $D=2$. $3\times 3$ wires (bottom left) are mapped into $3\times 3$ wires (top right) by layers of tensors that act along the $y$ and $x$ directions [some stages involve $3\times 4$ wires]. The generalization to $D>2$ is straightforward.

Figure 4

Left: Tensor network for ${\rho }^{[s]}={\mathrm{tr}}_{\mathcal{L}\setminus s}|\Psi ⟩⟨\Psi |$ in Lemma 1. Right: The gates outside the causal cone of $s$ can be removed by using the unitarity constraints of Eqs. (2, 3). The resulting tensor network can now be contracted by sequentially computing $O(\log N)$ density matrices $\{{\sigma }_1,{\sigma }_2,\dots \}$.

Figure 5

Efficient computation of ${\rho }^{[s]}$, Lemma 1. (i) Part of the causal cone ${\mathcal{C}}^{[s]}$ of ${\rho }^{[s]}$. (ii) Sequence of density matrices ${\sigma }_{2k-1}\to {\sigma }_{2k}\to {\sigma }_{2k+1}$ represented as tensors with 6 or 8 indices. (iii) ${\sigma }_{2k}$ expressed in terms of ${\sigma }_{2k-1}$ and isometries $w$. (iv) ${\sigma }_{2k+1}$ expressed in terms of ${\sigma }_{2k}$ and disentanglers $u$. By contracting all of the indices shared by two tensors in (iii) and (iv), ${\sigma }_{2k}$ and ${\sigma }_{2k+1}$ are obtained with cost $O(m^{p_1})$.

Figure 6

Renormalization group transformation associated to the MERA. The tensors in $\mathcal{M}$ define a sequence $\{{\mathcal{L}}_0,{\mathcal{L}}_1,\dots \}$ of increasingly coarser lattices in states $\{|{\Psi }_0⟩,|{\Psi }_1⟩,\dots \}$.

Figure 7

Causal cone ${\mathcal{C}}^{[s_1{s}_2]}$ for $D=1$. A small square at the end of an open wire indicates a trace over the corresponding degrees of freedom. The states ${\rho }^{[s_1{s}_2]}$ are obtained from ${\rho }_{\overline{\tau }}$ through a sequence of $O(\log r)$ transformations. Similar constructions hold for $D>1$.